Question

A charged particle (electron or proton) is introduced at the origin (𝑥 = 0, 𝑦 = 0, 𝑧 = 0) with a given initial velocity $\vec{v}$. A uniform electric field $\vec{E}$ and a uniform magnetic field $\vec{B}$ exist everywhere. The velocity $\vec{v}$, electric field $\vec{E}$ and magnetic field $\vec{B}$ are given in columns 1, 2 and 3, respectively. The quantities $𝐸_0 , 𝐵_0$ are positive in magnitude.

Column 1	Column 2	Column 3
(I) Electron with $\vec{v}=2 \frac{E_{0}}{B_{0}}\hat{x}$	(i) $\vec{E}=-E_{0}\hat{z}$	(P) $\vec{B}=-B_{0}\hat{x}$
(II)Electron with $\vec{v}= \frac{E_{0}}{B_{0}}\hat{y}$	(ii) $\vec{E}=-E_{0}\hat{y}$	(Q) $\vec{B}=-B_{0}\hat{x}$
(III) Proton with $\vec{v}=0$	(iii) $\vec{E}=-E_{0}\hat{x}$	(R) $\vec{B}=-B_{0}\hat{y}$
(IV)Proton with $\vec{v}=2 \frac{E_{0}}{B_{0}}\hat{x}$	(iv) $\vec{E}=-E_{0}\hat{x}$	(S) $\vec{B}=-B_{0}\hat{z}$

In which case will the particle describe a helical path with axis along the positive 𝑧 direction?

• A. (IV) (i) (S)
• B. (II) (ii) (R)
• C. (III) (iii) (P)
• D. (IV) (ii) (R)

Answer 1

Answer: (A)

In option (D), particle moves in a circle in $x - y$ plane without the electric field. With $\vec{ E }$ it is lifted up (along the z direction) with constant acceleration generating a helix with constant radius and increasing pitch.